Articles of polyhedra

Is an unit-cube polyhedron? What about other platonic solids?

Definitions According to my linear programming course and screenshot here (Finnish), a polyhedron is such that it can be constrained by a finite amount of inequalities such that $$P=\{\bar x\in \mathbb R^n | A \bar x\geq \bar b\}, A\in \mathbb R^{m\times n},\bar b\in \mathbb R^m$$ and a convex function $f(x)$ must satisfy $$f(\lambda \bar x+(1-\lambda)\bar […]

Geodesics on a polyhedron

Which sequences of adjacent edges of a polyhedron could be considered to be a geodesic? The edges of a face most surely will not, but the “equator” of the octahedron eventually will. But for what reasons? How do the defining property of a geodesic – having zero geodesic curvature – apply to a sequence of […]

Can a tetrahedron lying completely inside another tetrahedron have a larger sum of edge lengths?

Find 2 tetrahedrons $ABCD$ and $EFGH$ such that $EFGH$ lies completely inside $ABCD$. The sum of edge lengths of $EFGH$ is strictly greater than the sum of edge lengths of $ABCD$. I am completely stumped on this. Seems very counter intuitive to begin. I now have doubts if a solution exists or not. Source : […]

What is circumradius $R$ of the great disnub dirhombidodecahedron, or Skilling's figure?

The vertices of a uniform polyhedron all lie on a sphere. Out of curiosity, I looked at the circumradius $R$ of the $75$ polyhedra (non-prism) in the list (which assumed side $a=1$). For irrational $R$, almost all were roots of quadratics, quartics, and a few sextics that can factor over a square root: $\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7},\sqrt{11}$ (and […]

Name of this convex polyhedron?

Does anyone recognize / know the name of the convex polyhedron depicted below as the intersection of a Cuboctahedron and a Rhombicdodecahedron? Please note you have to interpret this picture and extract the intersection. Its faces are squares, equilateral triangles, and rectangles. The presence of rectangles takes it out of the realm of the Archimedean […]

Polyhedra from number fields

A question on the disnub mentions golden ($x^2-x-1=0$) gives the dodecahedron + much more. tribonacci ($x^3-x^2-x-1=0$) gives the snub cube. plastic ($x^3-x-1=0$) gives the snub icosidodecadodecahedron. All of these polyhedra can be built by generating 3D points in root $R$’s number with values $a + b R^c$ for integers $a$, $b$, $c$. After that, pick […]

Maximal unit lengths in 3D with $n$ points.

Given $n$ points in 3D space (V), what is the maximal number of unit distance lengths (E) between those points? Here are a few possibilities. Some of them are chromatic spindles. A collection of these best known configurations has been placed at Maximal Unit Lengths in 3D. V — E — figure 4 — 6 […]

How to cut a cube into an icosahedron?

Edit: Originally I asked this about a using a cube, but it is not a requirement to start with a cube, just how to end up with an icosahedron as on of the answers showed how to make dodecahedron a having started ith any shape. I have a cube, and I need to cut/file it […]

Maximum area of triangle inside a convex polygon

Prove that within any convex polygon of area $A$, there exists a triangle with area at least $cA$, where $c=\tfrac{3}{8}$. Are there any better constants $c$? I’m not sure how to approach this problem. It is easily proven that such a triangle should have its vertices on the perimeter of the polygon, but I don’t […]

Possible all-Pentagon Polyhedra

If a polyhedron is made only of pentagons and hexagons, how many pentagons can it contain? With the assumption of three polygons per vertex, one can prove there are 12 pentagons. Let’s not make that assumption, and only use pentagons. 12 pentagons: dodecahedron and tetartoid. 24 pentagons: pentagonal icositetrahedron. 60 pentagons: pentagonal hexecontahedron. 72 pentagons: […]